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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-2uplex | Structured version Visualization version GIF version |
Description: A couple is a set if and only if its coordinates are sets. (Contributed by BJ, 6-Oct-2018.) |
Ref | Expression |
---|---|
bj-2uplex | ⊢ (⦅𝐴, 𝐵⦆ ∈ V ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-pr21val 32194 | . . . 4 ⊢ pr1 ⦅𝐴, 𝐵⦆ = 𝐴 | |
2 | bj-pr1ex 32187 | . . . 4 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → pr1 ⦅𝐴, 𝐵⦆ ∈ V) | |
3 | 1, 2 | syl5eqelr 2693 | . . 3 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → 𝐴 ∈ V) |
4 | bj-pr22val 32200 | . . . 4 ⊢ pr2 ⦅𝐴, 𝐵⦆ = 𝐵 | |
5 | bj-pr2ex 32201 | . . . 4 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → pr2 ⦅𝐴, 𝐵⦆ ∈ V) | |
6 | 4, 5 | syl5eqelr 2693 | . . 3 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → 𝐵 ∈ V) |
7 | 3, 6 | jca 553 | . 2 ⊢ (⦅𝐴, 𝐵⦆ ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
8 | df-bj-2upl 32192 | . . 3 ⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐵)) | |
9 | bj-1uplex 32189 | . . . . 5 ⊢ (⦅𝐴⦆ ∈ V ↔ 𝐴 ∈ V) | |
10 | 9 | biimpri 217 | . . . 4 ⊢ (𝐴 ∈ V → ⦅𝐴⦆ ∈ V) |
11 | snex 4835 | . . . . 5 ⊢ {1𝑜} ∈ V | |
12 | bj-xtagex 32170 | . . . . 5 ⊢ ({1𝑜} ∈ V → (𝐵 ∈ V → ({1𝑜} × tag 𝐵) ∈ V)) | |
13 | 11, 12 | ax-mp 5 | . . . 4 ⊢ (𝐵 ∈ V → ({1𝑜} × tag 𝐵) ∈ V) |
14 | unexg 6857 | . . . 4 ⊢ ((⦅𝐴⦆ ∈ V ∧ ({1𝑜} × tag 𝐵) ∈ V) → (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐵)) ∈ V) | |
15 | 10, 13, 14 | syl2an 493 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐵)) ∈ V) |
16 | 8, 15 | syl5eqel 2692 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⦅𝐴, 𝐵⦆ ∈ V) |
17 | 7, 16 | impbii 198 | 1 ⊢ (⦅𝐴, 𝐵⦆ ∈ V ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∈ wcel 1977 Vcvv 3173 ∪ cun 3538 {csn 4125 × cxp 5036 1𝑜c1o 7440 tag bj-ctag 32155 ⦅bj-c1upl 32178 pr1 bj-cpr1 32181 ⦅bj-c2uple 32191 pr2 bj-cpr2 32195 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-suc 5646 df-1o 7447 df-bj-sngl 32147 df-bj-tag 32156 df-bj-proj 32172 df-bj-1upl 32179 df-bj-pr1 32182 df-bj-2upl 32192 df-bj-pr2 32196 |
This theorem is referenced by: (None) |
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